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Vol. 6 (2001) Paper no. 13, pages 1–27
Journal URL
http://www.math.washington.edu/~ejpecp/
Paper URL
http://www.math.washington.edu/~ejpecp/EjpVol6/paper13.abs.html
BOUNDARY CONDITIONS FOR ONE-DIMENSIONAL
BIHARMONIC PSEUDO PROCESS
Kunio Nishioka
Department of Mathematics, Tokyo Metropolitan University
Minami-Ohsawa, Hachioji-shi, Tokyo 192-0397, Japan
kunio@comp.metro-u.ac.jp
Abstract We study boundary conditions for a stochastic pseudo processes corresponding to
the biharmonic operator. The biharmonic pseudo process (BP P for short). is composed, in a
sense, of two different particles, a monopole and a dipole. We show how an initial-boundary
problems for a 4-th order parabolic differential equation can be represented by BP P with various
boundary conditions for the two particles: killing, reflection and stopping.
Keywords Boundary conditions for biharmonic pseudo process, killing, reflection, stopping
AMS Subject Classification Primary. 60J50; Secondary. 35K35, 60G18, 60G20, 60J30.
Submitted to EJP on September 6, 2000. Final version accepted on May 21, 2001.
1
Introduction
We will call −42 ≡ −∂x4 a biharmonic operator. It plays an important role in the theory of
elasticity and fluid dynamics. For instance the parabolic differential equation
∂t u(t, x) = −∂x4 u(t, x),
t > 0,
x ∈ R1 ,
(1.1)
is closely related to the Kuramoto-Sivashinsky equation and the Cahne-Hilliard equation (see
[17, Chapter III §4]). It is easy to see that the fundamental solution p(t, x) to (1.1) is given by
Z ∞
1
p(t, x) =
dξ exp{−iξx − ξ 4 t}, t > 0, x ∈ R 1 .
(1.2)
2π −∞
This p(t, x) takes negative values (see Hochberg [8]), so no stochastic process corresponds to
(1.1) in the usual sense.
Several attempts have been made to relate the biharmonic operator to random processes. Krylov
[9] considered a stochastic pseudo process whose “transition probability density” was p(t, x) as
in (1.2), despite its taking negative values. We will call this pseudo process a biharmonic pseudo
process or BP P for short (it is occasionally called the Krylov motion).
Following Krylov’s idea, Hochberg [8] started a systematic study of BP P . His article included
a definition of a stochastic integral with respect to BP P . Nishioka [14] calculated the joint
distribution of the first hitting time and place for BP P hitting the boundary of a half-line. This
was extended by Nakajima and Sato [10] to the space-time case. Some new developments in
this direction can be found in a paper by Beghin, Hochberg and Orsingher [1].
Other attempts included a paper by Funaki [7] who introduced the concept of iterated Brownian
motion. After his pioneering work, Burdzy [2] began to study path properties of iterated Brownian motion; a large number of related papers followed. Burdzy and Ma̧drecki [3, 4] modified
Funaki’s model in a different way; they defined a stochastic integral with respect to the their
process.
This paper is devoted to the process BP P on a half-line. We will study various boundary
conditions for the probabilistic model and relate them to their analytic counterparts. We will
motivate our results by first reviewing the classical case of the standard Brownian motion on
(0, ∞).
Consider the following initial-boundary value problem for the heat equation:
∂t u(t, x) = (1/2) ∂x2 u(t, x), t > 0, x > 0;
∂x` u(t, 0)
= 0,
u(0, x) = f (x), x > 0;
t > 0,
(1.3)
where ` may take values 0, 1 or 2. Solutions to this initial-boundary value problem may be represented using Brownian motion on (0, ∞) which is killed, reflected, or stopped at the boundary
point 0, depending on the value of `. The relationship can be summarized as follows:
`
0
1
2
Table 1.1.
Brownian boundary condition
killing
reflection
stopping
2
We note that there is no analytic difference between Brownian transition probabilities on the
open interval (0, ∞) in the cases when the Brownian particle is killed or stopped at the boundary
but we make the distinction in Table 1.1 to emphasize the analogy with some results on BP P .
In this article, we will establish a relationship analogous to Table 1.1, between BP P and several
initial-boundary value problems for (1.1).
We start with a few heuristic ideas. A big difference between a Brownian motion and BP P ,
proved in [13], is that the Brownian motion is a model for the motion of a single particle but when
the BP P leaves the interval (0, ∞), it appears to consist of two types of particles. We will call
these particles a monopole and a dipole. The names are justified by a result on the conservation
of charges given in the last section of this article. The particles behave independently at the
boundary. Therefore we need two different boundary conditions in order to solve the initialboundary value problem for (1.1). In a sense, each boundary condition controls the behavior
of a monopole or a dipole, although the exact relationship, summarized in Table 1.2 below, is
more complicated than that.
We proceed with a more formal presentation of the main results although the fully rigorous
statements are postponed until later in the paper. Consider the following initial-boundary value
problems for (1.1),
∂t v(t, x) = −∂x4 v(t, x), t > 0, x > 0;
∂x` v(t, 0)
= 0 and
∂xm v(t, 0)
v(0, x) = f (x), x > 0;
= 0, t > 0,
(1.4)
where f is a given bounded smooth function on [0, ∞) and ` and m can take the values ` =
0, . . . , 4 and m = 0, . . . , 5. The following two tables summarize our main results.
`=0
`=1
`=2
`=3
`=4
m=0
−−
−−
−−
−−
−−
m=1
00
−−
−−
−−
−−
Table 1.2.
m=2 m=3
0r
H
r0
H
−−
H
−−
−−
−−
−−
m=4
×
s0
sr
H
−−
m=5
0s
×
H
H
ss
For some values of ` and m, the initial boundary value problem has a probabilistic representation
using BP P with killing, reflection or stopping for the monopole and dipole. The correspondence
between the various values of ` and m and boundary conditions for BP P is coded in Tables 1.2
and 1.3 in the self-evident manner.
killing
for monopole
reflection
for monopole
stopping
for monopole
Table 1.3.
killing for dipole reflection for dipole
00
0r
(see §4)
(see §6)
r0
−−
(see §5.1)
(see §5.3)
s0
sr
(see §7.2)
(see §7.3)
3
stopping for dipole
0s
(see §8.1)
−−
(see §8.2)
ss
(see §7.1)
In Table 1.2, the symbol “×” means that (1.4) is not well-posed with such analytic boundary
conditions (see Example 7.9). The symbol “H” means such analytic boundary conditions are not
related to BP P with killing, reflection or stopping at the boundary. However, these analytic
conditions can be related to BP P . This requires a new idea of a “higher order reflecting
boundary” which will be discussed in a forthcoming article. The diagonal entries in Table 1.2
and those below the diagonal are irrelevant for obvious reasons and so they are marked with
“−−”. The same symbol in Table 1.3 means that we have not found a relationship between this
set of analytic conditions and BP P .
In §2, we review results from [13, 14] on the joint distribution of the first hitting time and place
for BP P and the concepts of a monopole and dipole. In §3 through §8, we construct and study
BP P ’s with various boundary conditions.
The construction of the BP P in the case when we have killing or stopping of the monopole at
the boundary is similar in a sense to the Brownian motion case. Only slight modifications are
needed when the same boundary conditions are prescribed to the dipole.
When we set the reflecting boundary for a monopole or dipole, we cannot apply a direct analogy
with the reflecting Brownian motion (see §5.2). We will construct a BP P with the reflection as
the limit of a suitable sequence of approximations.
In §9, we verify the law of the conservation of charges, and conclude that the total amount of
charges is conservative if and only if there is no killing for a monopole.
We will use some basic results on solutions of 4-th order PDE’s, Laplace and Fourier-Laplace
transforms through the paper. These can be found in [5, 6, 16] for instance.
Acknowlegement The author would like to thank the referee made many important and
precise suggestions for improvements of the paper.
2
Preliminaries
We start with a review of several function spaces used in the article.
Let X denote either the one dimensional Euclidean space R 1 or the half-line [0, ∞). Bb (X) will
stand for the space of all bounded measurable functions defined on X. C(X) will be the space of
all continuous functions on X while Cb (X) will be the space of all bounded functions in C(X).
C1 (X) will denote the space of all continuously differentiable functions on X and C1b (X) will
be the space of all bounded functions in C1 (X).
D[0, ∞) will denote the space of all right continuous functions defined on [0, ∞) which have left
hand limits.
S will be the space of all Schwartz class functions defined on
R1 .
Dirac’s delta function will be denoted by δ(x) and δa (db) will be the delta measure with the unit
mass at a point {a}, so we may use the convention δa (db) = δ(a − b) db.
For a fixed positive t, the function p(t, x) of (1.2) is an even function of x and it belongs to S.
4
For positive t and s, the following formulae hold
Z ∞
x dx p(t, x) = 1, p(t, x) = t−1/4 p 1, 1/4 ,
t
−∞
Z ∞
and
dy p(t, y − x) p(s, y) = p(t + s, x).
(2.1)
−∞
Note that p(t, x) can take negative values. Hochberg [8] proved that
p(1, |x|) = a|x|−1/3 exp{−b|x|4/3 } cos c|x|4/3 + a lower order term
for large |x|, where a, b, and c are positive constants. It follows that
Z ∞
Z ∞
|p(t, x)| dx =
|p(1, x)| dx = a constant > 1 for any t > 0.
−∞
(2.2)
−∞
e x on cylinder sets in
Using this p(t, x), Krylov [9] defined a finitely additive signed measure P
[0,∞)
[0,∞)
R
. A cylinder set in R
, say Γ, is a set such that
Γ = {ω ∈ R [0,∞) : ω(t1 ) ∈ B1 , · · · , ω(tn ) ∈ Bn }
where 0 ≤ t1 < · · · < tn and Bk ’s are Borel sets in R 1 . We put
Z
Z
e x [Γ] ≡
P
dy1 · · ·
dyn p(t1 , y1 − x)×
B1
(2.3)
(2.4)
Bn
× p(t2 − t1 , y1 − y2 ) · · · p(tn − tn−1 , yn − yn−1 ).
e x is a σ–additive finite measure on R n , but it is not a
If we fix 0 ≤ t1 < · · · < tn , then this P
σ–additive measure on the smallest σ–field which includes all cylinder sets in R [0,∞) .
We say that a function f defined on
R[0,∞)
is tame, if
f (ω) = g(ω(t1 ), · · · , ω(tn )),
ω ∈ R [0,∞) ,
(2.5)
for a Borel function g on R n and 0 ≤ t1 < · · · < tn . For each tame function f , we define its
expectation in the usual way—if f is as in (2.5), then we set
Z
e
e x [dω(t1 ) × · · · × dω(tn )],
Ex [f (ω)] = f (ω) P
(2.6)
if the right hand side exists.
We extend the expectation to other functions as follows. Let n and N be natural numbers. For
each ω ∈ R[0,∞) , we set

if k/2n ≤ t < (k + 1)/2n and t < N
 ω(k/2n )
ωnN (t) ≡

ω(N )
if t ≥ N .
Definition 2.1. We say that a function F defined on
following conditions,
5
R[0,∞)
is admissible if F satisfies the
(i) for each n and N , F (ωnN ) is tame,
(ii) for each ω ∈ R[0,∞) , limn→∞ limN →∞ F (ωnN ) = F (ω),
P
k
k−1
e
e
(iii) there exists limn→∞ limN →∞ N
k=1 |Ex [F (ωn )] − Ex [F (ωn )]|.
We define the expectation of an admissible function F (ω) by
e x [F (ω N )].
Ex [F (ω)] ≡ lim lim E
n
n→∞ N →∞
The expectation is unique if it exists, due to (iii) of Definition 2.1 (see [14]). Let A be a subset
in R [0,∞) . When the characteristic function IA (ω) is admissible, we let
Px [A] ≡ Ex [IA (ω)].
(2.7)
For a positive number α, Hα [0, ∞) will denote the space of all functions defined on [0, ∞) which
satisfy Hölder’s condition of order α. According to Krylov [9],
Px –total variation of Hα [0, ∞) c = 0
if
α < 14 .
This implies that the total mass of |Px | is concentrated on C[0, ∞). However for a technical
reason related to Definition 2.1 (we need to deal with ωnN ), we will take a larger space D[0, ∞)
as the path space of the pseudo process corresponding to Px . From now on, we will identify Px
and Ex with their restrictions to D[0, ∞).
Definition 2.2. A biharmonic pseudo process, or BP P in short, is the family of finitely additive
signed measures {Px : x ∈ R1 }, defined in (2.4) and (2.7). The domain of Px is a finitely additive
field in D[0, ∞) which includes all cylinder sets.
3
The first hitting time distribution, monopole and dipole
We will review results from [13, 14] in this section..
3.1
The distribution of the first hitting time and place
Given ω ∈ D[0, ∞), the random time
τ0 (ω) ≡ inf{t > 0 : ω(t) < 0},
is called the first hitting time of the interval (−∞, 0). It can be proved that the function
exp{−λτ0 (ω) + iβω(τ0 )}
(3.1)
is admissible, and we can calculate its expectation. For each λ > 0 and β ∈ R 1 ,
Ex [exp{−λτ0 (ω) + iβω(τ0 )}]
1 = √ θ1 exp{λ1/4 θ2 x} + θ1 exp{λ1/4 θ2 x}
2
iβ + √
−i exp{λ1/4 θ2 x} + i exp{λ1/4 θ2 x} ,
1/4
2λ
6
(3.2)
where θ1 ≡ exp{πi/4} and θ2 ≡ exp{3πi/4}.
In the classical probability setting, one can derive the joint distribution of the first hitting time
and place
Px [τ0 (ω) ∈ dt, ω(τ0 ) ∈ da]
(3.3)
from (3.2), using Bochner’s theorem. However the theorem cannot be applied to BP P , since
(3.2) is not positive definite in β, and so we have to extend the notion of the “joint distribution”
itself.
Given functions φ and ϕ in S, it can be proved that the function
exp{−λτ0 (ω)} φ(τ0 (ω)) ϕ(ω(τ0 ))
is admissible and its expectation defines a continuous bilinear functional on S ×S. This fact is directly implied by Schwartz’s result on Fourier-Laplace transforms of his temperate distributions.
The following lemma is a slight modification of his result.
Lemma 3.1 ([15]). Let c be a positive number and U (λ, β) be a complex-valued function of
a real variable β and a complex variable λ with <λ > c. Assume that for each real β, U is a
holomorphic function in λ with <λ > c and it satisfies
k
|U (λ, β)| ≤ C 1 + |λ| + |β|
for <λ > c and β ∈ R 1 ,
(3.4)
where C and k are non-negative constants. Then U is the Fourier-Laplace transform of a
Schwartz temperate distribution whose support lies in [0, ∞) × R 1 .
We will always take the principal value for λ1/4 . If <λ > c for any positive number c, then
−π/8 < arg λ1/4 < π/8. This implies that (3.2) satisfies (3.4). Hence there exists a Schwartz
temperate distribution q(t, a; x) such that
Ex [exp{−λτ0 (ω)} φ(τ0 (ω)) ϕ(ω(τ0 ))]
Z ∞ Z
=
dt da exp{−λt} q(t, a; x) φ(t) ϕ(a),
λ > c > 0,
(3.5)
0
holds for all φ and ϕ in S.
In the classical probability theory, distributions of real-valued random variables may be thought
of as non-negative continuous linear functionals on the function space Cb (R 1 ). We define distributions of functions of BP P to be continuous linear functionals on a function space which
includes S.
Definition 3.2. We call Schwartz’s temperate distribution q(t, a; x) in (3.5) the density of the
distribution (3.3), and let
Px [τ0 (ω) ∈ dt, ω(τ0 ) ∈ da] ≡ q(t, a; x) dt da,
which is a continuous bilinear functional on S × S for each x ≥ 0.
Proposition 3.3 ([14]). Let x ≥ 0. Then in the distribution sense,
Px [τ0 (ω) ∈ dt, ω(τ0 ) ∈ da] = K(t, x) δ(a) − J(t, x) δ0 (a) dt da
7
(3.6)
where δ0 (a) is the first derivative of Dirac’s delta function δ(a),
Z
2 ∞
K(t, x) ≡
dy exp{−y 4 t} y 3 sin yx − cos yx + exp{−yx} ,
π 0
Z
2 ∞
J(t, x) ≡
dy exp{−y 4 t} y 2 sin yx − cos yx + exp{−yx} .
π 0
(3.7)
Moreover, the support of the distribution in (3.6) is [0, ∞) × {0}.
Remark 3.4 ([14]). (i) For each t > 0, K(t, x) and J(t, x) are C∞
b [0, ∞) functions in x. For
each x > 0, they are also C∞
[0,
∞)
functions
in
t.
b
(ii) For some positive constant C independent of t and x, we have
!n
C x2 t1/4
|K(t, x)| ≤
for n = 0, · · · , 6;
x
t3/2
!n
C x2 t1/4
|J(t, x)| ≤
for n = 0, · · · , 5.
x
t5/4
(iii) For x > 0,
Z
Z
∞
K(t, x) dt = 1
∞
and
0
J(t, x) dt = x.
0
(iv) Let f be a bounded continuous function on [0, ∞). For t > 0,
Z
lim
x→0 0
Z
t
ds K(s, x) f (s) = f (0) and
lim
x→0 0
t
ds J(s, x) f (s) = 0.
The explicit formula (3.6) makes it possible to extend (3.3) to a continuous bilinear functional
on a larger space than S × S.
Corollary 3.5 ([14]). The distribution (3.3) can be extended to a continuous bilinear functional
on Bb [0, ∞) × C1 (R 1 ).
The strong Markov property holds for BP P at time τ0 (ω).
Proposition 3.6 (Strong Markov Property [14]). Let y < 0 < x. The following holds in
the sense of continuous linear functionals on Bb [0, ∞),
Z
Px [ω(t) ∈ dy] =
3.2
t
Z
s=0
∞
a=−∞
Px [τ0 (ω) ∈ ds, ω(τ0 ) ∈ da] Pa [ω(t − s) ∈ dy].
(3.8)
Monopole and dipole
In physics a particle is called a dipole when it carries two charges of equal magnitude but opposite
signs. A small magnet is a typical example of a dipole. Heuristically, one may represent a dipole
by
1
1
− δ(a + ) +
δ(a − ).
(3.9)
2
2
8
As → 0, (3.9) converges to −δ0 (a) in the distribution sense. Therefore we will call −δ0 (a) a
dipole. For the Brownian motion B(t), it is well known that
x
Px [τ0 ∈ dt, B(τ0 ) ∈ da] = √
exp{−x2 /2t} δ(a) dt da; x > 0.
2πt3
Comparing this with (3.6), we see that in a sense, BP P behaves as a mixture of two particles
of different types when it hits an interval.
Informally speaking, a particle of the first type is represented by δ(a) and carries a charge of a
single sign just as a Brownian particle does. The second particle is represented by −δ0 (a) and
corresponds to a dipole. The following informal definition will enable us to present our results
using intuitive notation.
Definition 3.7. We will call a particle represented by δ(a) a monopole and a particle represented
by −δ0 (a) a dipole. We define distributions of the first hitting time and place for the monopole
and dipole as follows.
Px [τ0 (ω) ∈ dt, ω(τ0 ) is a monopole and in da] ≡ K(t, x) δ(a) dt da,
Px [τ0 (ω) ∈ dt, ω(τ0 ) is a dipole and in da] ≡ J(t, x) (−δ0 (a)) dt da,
where the first distribution is a continuous linear functional on Bb [0, ∞) × C(R1 ) and the latter
is a functional on Bb [0, ∞) × C1 (R 1 ).
Corollary 3.8. In the sense of continuous linear functionals on Bb [0, ∞) × C1 (R 1 ),
Pb [τ0 (ω) ∈ dt, ω(τ0 ) ∈ da]
= Px [τ0 (ω) ∈ dt, ω(τ0 ) is a monopole and in da]
(3.10)
+ Px [τ0 (ω) ∈ dt, ω(τ0 ) is a dipole and in da].
3.3
The initial particle
When BP P starts with an initial distribution µ(dy), its distribution at time t is given by
Z
µ(dy) Py [ω(t) ∈ db].
(3.11)
Recall from Definition 3.7 that δ(y) stands for the monopole and the dipole is represented by
−δ0 (y). Therefore when a monopole starts from a point x, the distribution of the process at
time t is
Z
dy δ(y − x) Py [ω(t) ∈ db] = p(t, b − x) db = Px [ω(t) ∈ db].
On the other hand, when a dipole starts from a point x, the distribution at time t is
Z
dy (−δ0 (y − x)) Py [ω(t) ∈ db] = ∂x p(t, x − b) db.
We have a similar formula for the joint distribution of the first hitting time and place. When a
monopole starts from a point x, the distribution is
Z
n
dy δ(y − x) Py [τ0 (ω) ∈ dt, ω(τ0 ) is a monopole and in da]
o
(3.12)
+ Py [τ0 (ω) ∈ dt, ω(τ0 ) is a dipole and in da ]
= same as the right hand side of (3.10).
9
On the other hand when a dipole starts from x, then the distribution is
Z
n
dy (−δ0 (y − x)) Py [τ0 (ω) ∈ dt, ω(τ0 ) is a monopole and in da ]
o
+ Py [τ0 (ω) ∈ dt, ω(τ0 ) is a dipole and in da ]
n ∂K
o
∂J
=
(t, x) δ(a) −
(t, x) δ0 (a) dt da.
∂x
∂x
(3.13)
Remark 3.9. Informally speaking, (3.13) shows that a dipole generates both a monopole and
a dipole at the hitting time of (−∞, 0), just like a monopole does.
3.4
Some Laplace transforms
For future reference, we list some Laplace and Fourier-Laplace transforms of K(t, x), J(t, x) and
p(t, x). Let λ > 0 and x ≥ 0. From Proposition 3.3, we have
Z ∞
K̂(λ, x) ≡
dt exp{−λt} K(t, x)
0
!
(3.14)
√
√
λ1/4 x π
1/4
√ −
= 2 exp{−λ x/ 2} cos
,
4
2
Z
ˆ x) ≡
J(λ,
∞
dt exp{−λt} J(t, x)
0
√
√
2
λ1/4 x
= 1/4 exp{−λ1/4 x/ 2} sin √ ,
λ
2
Z ∞ Z ∞
Ĝ(λ, x, β) ≡
dt
dy exp{−λt + iβy} Px [ω(t) ∈ dy]
0
−∞
exp{iβx}
=
.
λ + β4
In addition, we define a new function
Z ∞ Z
00
Ĝ (λ, x, β) ≡
dt
exp{−λt + iβb}Px [ω(t) ∈ db]−
0
b∈R
Z ∞ Z
−
dt
exp{−λt + iβb}×
0
b∈R
Z t Z
n
×
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da]
s=0 a∈R
o
+ Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da] Pa [ω(t − s) ∈ db]
1 ˆ(λ, x) .
= Ĝ(λ, x, β) −
K̂(λ,
x)
+
iβ
J
λ + β4
(3.15)
(3.16)
(3.17)
It is elementary to check that
K̂(λ, 0) = 1,
∂x3 K̂(λ, 0) =
∂x K̂(λ, 0) = 0,
√ 3/4
2λ ,
√
∂x2 K̂(λ, 0) = − λ,
∂x4 K̂(λ, 0) = −λ,
10
∂x5 K̂(λ, 0) = 0;
(3.18)
√
ˆ 0) = 0, ∂x J(λ,
ˆ 0) = 1, ∂ 2 J(λ,
ˆ 0) = − 2λ1/4 ,
J(λ,
x
√
ˆ 0) = 0, ∂ 5 J(λ,
ˆ 0) = −λ;
∂ 3 Jˆ(λ, 0) = λ, ∂ 4 J(λ,
x
x
(3.19)
x
Ĝ00 (λ, 0, β) = 0,
∂x Ĝ00 (λ, 0, β) = 0,
√
√
−β 2 + λ + i 2λ1/4 β
2 00
∂x Ĝ (λ, 0, β) =
λ + β4
√
√ 3/4
− 2λ − iβ 3 − i λβ
3 00
∂x Ĝ (λ, 0, β) =
,
λ + β4
∂x4 Ĝ00 (λ, 0, β) = 1,
4
(3.20)
∂x5 Ĝ00 (λ, 0, β) = iβ.
Killing boundaries for both particles
Definition 4.1. Let
P00
x [ω(t) ∈ db] = Px [ω(t) ∈ db]−
Z t Z
n
−
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da]
s=0 a∈R
o
+ Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da] Pa [ω(t − s) ∈ db].
(4.1)
We will refer to {P00
x [ω(t) ∈ db] : x ≥ 0} as a BP P with killing boundaries for both particles.
Remark 4.2. Here is an intuitive description of BP P with killing boundaries for both particles.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. After BP P hits (−∞, 0), both the monopole and dipole are killed.
Theorem 4.3. (i) For x ≥ 0, b ≥ 0, and t > 0, we define a function
p00 (t, x, b) ≡ p(t, b − x) −
Then, we have
Z
Z
∞
∞
dt
0
Z
t
0
ds K(s, x) p(t − s, b) + J(s, x) ∂b p(t − s, b) .
(4.2)
db exp{−λt + iβb) p00 (t, x, b) = Ĝ00 (λ, x, β).
−∞
(ii) The following linear functionals are identical and continuous on Bb [0, ∞),
00
P00
x [ω(t) ∈ db] = p (t, x, b) db.
(4.3)
Proof. We easily obtain (4.2) and (4.3) when we recall the definitions of the terms on the right
hand side of (4.1) from Section 3.2. It is straightforward to check that (3.20) is the FourierLaplace transform of (4.2) in the usual sense. Part (ii) follows from the fact that p(t, x) ∈ S
and from the estimates stated in Remark 3.4. 11
Remark 4.4. The following assertion easily follow from (4.2).
(i) If f is a bounded continuous function on [0, ∞), then
Z ∞
lim
P00
x [ω(t) ∈ db] f (b) = f (x).
t→0 0
(ii) If x = 0, P00
x [ω(t) ∈ A] = 0 for any Borel set A ∈ (0, ∞).
As expected, the distribution P00
x [ω(t) ∈ db] solves (1.4) with the Dirichlet boundary condition.
Theorem 4.5 ([14]). Let f be a bounded smooth function on the interval [0, ∞) such that
f (0) = 0 = f 0 (0). We put
Z ∞
v(t, x) ≡
P00
(4.4)
x [ω(t) ∈ db] f (b).
0
Then in the classical sense, this v satisfies (1.4) with ` = 0 and m = 1.
Corollary 4.6. P00
x [ω(t) ∈ db] satisfies the Chapman-Kolmogorov equations, that is
Z ∞
db p00 (t, x, b) p00 (s, b, y) = p00 (t + s, x, y).
(4.5)
0
Proof. Fix s > 0 and y ≥ 0. Theorem 4.5 asserts that
Z ∞
v(t, x) ≡
db p00 (t, x, b) p00 (s, b, y)
0
is a solution of (1.4) where
f (x) ≡ p00 (s, x, y),
` = 0 and m = 1. The function ṽ(t, x) ≡ p00 (t + s, x, y) is also a solution of (1.4). It is well
known that the solution is unique so it follows that v = ṽ. The corollary is proved. Remark 4.7. Due to Corollary 4.6, we may consider P00
x [ω(t) ∈ db] of (4.3) as a finitely
additive signed measure on cylinder sets in D[0, ∞) in the same way as in (2.4).
5
5.1
Reflecting boundary for the monopole
Reflection for the monopole and killing of the dipole
The construction of the process BP P with the boundary conditions specified above will use an
approximating sequence. We will prove existence of a family { Pr0
x [ω(t) ∈ db] : x ≥ 0} satisfying
0
Pr0
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da] Pr0
+a [ω(t − s) ∈ db].
0
(5.1)
−∞
Remark 5.1. Heuristically, { Pr0
x [ω(t) ∈ db] : x ≥ 0} represents the following process. Fix
> 0.
12
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it restarts from the point + ω(τ0 ).
3. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it is killed.
Assume for the moment that Pr0
x [ω(t) ∈ db] is a Schwartz temperate distribution and denote
its Fourier-Laplace transform by
Z ∞ Z
r0
r0
U (λ, x, β) ≡
dt
Px [ω(t) ∈ db] exp{−λt + iβb}].
0
Apply the Fourier-Laplace transform to the both sides of (5.1) to obtain
U r0 (λ, x, β) = Ĝ00 (λ, x, β) + K̂(λ, x) U r0 (λ, , β),
x > 0.
(5.2)
Lemma 5.2. (i) When > 0, the following is a solution of (5.2):
U r0 (λ, x, β) = Ĝ00 (λ, x, β) + K̂(λ, x)
Ĝ00 (λ, , β)
1 − K̂(λ, )
.
(5.3)
(ii) For small > 0, this U r0 satisfies (3.4) for any c > 0, and it is the Fourier-Laplace
transform of a Schwartz temperate distribution Pr0
x [ω(t) ∈ db].
Proof. (i) Put x = in (5.2) to see that
U r0 (λ, , β) = Ĝ00 (λ, , β) + K̂(λ, ) U r0 (λ, , β).
This linear equation is easily solved with respect to U r0 . This and (5.2) yield (5.3).
(ii) Let <λ > c > 0. We take the principal value for λ1/4 and recall that −π/8 < arg λ1/4 < π/8.
After applying this fact to (3.14) through (3.17), we see that (3.4) holds for K̂, Jˆ, and Ĝ00 , and
also for the fraction on the right hand side of (5.3) when > 0 is small. Now Lemma 3.1 implies
our assertion. Lemma 5.3. In the distribution sense, Pr0
x [ω(t) ∈ db] converges to a limit as → 0.
Definition 5.4. This above limit will be denoted by {Pr0
x [ω(t) ∈ db] : x ≥ 0} and called the
distribution of BP P with the reflecting boundary for the monopole and the killing boundary for
the dipole.
Proof of Lemma 5.3. Recalling (3.18) through (3.20), we let → 0 in (5.3), and obtain
U r0 (λ, x, β) ≡ lim U r0 (λ, x, β)
→0
= Ĝ00 (λ, x, β) − K̂(λ, x)
∂x2 Ĝ00 (λ, 0, β)
∂x2 K̂(λ, 0)
(5.4)
.
Now U r0 clearly satisfies (3.4), and so it is the Fourier-Laplace transform of a Schwartz temperate
distribution, that is Pr0
x [ω(t) ∈ db].
13
Assuming that <λ > 0, we calculate the usual inverse Fourier-Laplace transform of
∂x2 Ĝ00 (λ, 0, β)/∂x2 K̂(λ, 0). Let b ≥ 0. By the residue theorem, we have
Z L
∂ 2 Ĝ00 (λ, 0, β) 1
r0
I (λ, b) ≡
dβ exp{−iβb} − x
lim
2π L→∞ −L
∂x2 K̂(λ, 0)
(5.5)
i
1/4
1/4
=√
exp{λ θ2 b} − exp{λ θ2 b} ,
2λ3/4
where θ2 ≡ exp{i3π/4} and θ2 is its complex conjugate. Applying the usual inverse Laplace
transform to this I r0 , we obtain
Z 1+iM
1
r0
Q (t, b) ≡
dλ exp{λt} I r0 (λ, b)
lim
2πi M →∞ 1−iM
(5.6)
Z
2 ∞
−y 4 t
=
cos yb + sin yb − exp{−yb} ,
dy e
π 0
which is a smooth bounded function in b ≥ 0 if t > 0.
Theorem 5.5. For x ≥ 0, b ≥ 0, and t > 0, we define a function
Z t
pr0 (t, x, b) ≡ p00 (t, x, b) +
ds K(s, x) Qr0 (t − s, b),
(5.7)
0
where p00 is the function in (4.2) and Qr0 is defined in (5.6). Then we have
r0
Pr0
x [ω(t) ∈ db] = p (t, x, b) db,
(5.8)
both linear functionals in (5.8) are continuous on Bb [0, ∞) and
Z ∞
Pr0
x [ω(t) ∈ db] = 1.
0
Proof. The first part of the theorem follows from (5.4), (5.6), and Lemma 5.3. From (5.4) with
(3.18) through (3.20), we deduce that U r0 (λ, x, 0) = 1/λ, and this justifies the last assertion.
The distribution Pr0
x is related to the PDE in (1.4) with the Neumann boundary conditions.
Theorem 5.6. (i) Let f be a bounded smooth function on the interval [0, ∞) such that f 0 (0) =
0 = f 00 (0). We put
Z ∞
v(t, x) ≡
Pr0
(5.9)
x [ω(t) ∈ db] f (b).
0
Then in the classical sense, this v satisfies (1.4) with ` = 1 and m = 2.
(ii) Pr0
x [ω(t) ∈ db] satisfies Chapman-Kolmogorov equations (4.5), and Remark 4.7 holds with
r0
Px [ω(t) ∈ db] instead of P00
x [ω(t) ∈ db].
Proof. Recall (3.18) through (3.20). From (5.4), we have
∂x4 U r0 (λ, x, β) + λU r0 (λ, x, β) = exp{iβ x} for ∀x > 0,
and
∂x U r0 (λ, 0, β) = 0
(5.10)
and ∂x2 U r0 (λ, 0, β) = 0.
Since the Fourier-Laplace transform is unique for our pr0 , these prove the first part of the
theorem. The second part is immediate from the same argument as in the proof of Corollary
4.6. 14
5.2
An analogy to the classical reflecting Brownian motion
For the Brownian motion B(t), it is well known that the following two process X1 and X2 have
the same distributions. They are referred to as reflecting Brownian motions. Let τ0 be the first
hitting time of the point x = 0.
X1 (t) ≡ |B(t)|,
X2 (t) ≡
B(t)
B(t) − inf τ0 ≤s≤t B(s)
if t < τ0 ,
if t ≥ τ0 .
It is not difficult to check that similarly defined transformations of BP P do not have identical
distributions. Keeping this fact in mind, we define a new BP P as follows.
Definition 5.7. Let {Pm
x [ω(t) ∈ db] : x ≥ 0} be defined by
00
Pm
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da]×
0
−∞
(5.11)
× Pa [ω(t − s) − inf ω(t) ∈ db].
u≤t−s
Remark 5.8. We present an intuitive view of the last definition.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it is killed.
3. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it restarts from the point x = 0, and
moves according to the distribution of [ω(t + τ0 ) − inf τ0 ≤s≤t+τ0 ω(s)].
In [14], the following was proved. For b ≥ 0,
P0 [ω(t) − inf ω(s) ∈ db] = Qr0 (t, b) db,
0≤s≤t
where Qr0 is given in (5.6). We record this partial analogy with the classical reflected Brownian
motion in the next theorem.
m
Theorem 5.9. The distributions Pr0
x [ω(t) ∈ db] in Theorem 5.5 and Px [ω(t) ∈ db] in (5.11)
are identical.
Proof. The result follows easily from (5.11) and Theorem 5.5.
5.3
Reflecting boundaries for both particles
The construction of BP P with these boundary conditions requires an approximation procedure
similar to that in the previous section. We will prove that there exists a family { Prr
x [ω(t) ∈
15
db] : x ≥ 0} satisfying the following equation.
00
Prr
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z t Z ∞ n
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da]
s=0 a=−∞
o
+ Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da ] Prr
+a [ω(t − s) ∈ db].
(5.12)
Remark 5.10. Heuristically, we are dealing with the following process. Let be a fixed positive
number.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. After BP P hits (−∞, 0), it restarts from the point + ω(τ0 ) irrespective of whether ω(τ0 )
is a monopole or a dipole.
rr its Fourier-Laplace
Assuming for the moment existence of Prr
x [ω(t) ∈ db], we denote by U
transform. Recall (3.13) and note that (5.12) is transformed into the following equation. For
x > 0,
U rr (λ, x, β) = Ĝ00 (λ, x, β)+
+ K̂(λ, x) U rr (λ, , β) + Jˆ(λ, x) ∂x U rr (λ, , β).
(5.13)
Lemma 5.11. (i) For > 0, the following is a solution of (5.13):
U rr (λ, x, β) = Ĝ00 (λ, x, β)
ˆ λ)) Ĝ00 (λ, , β) + J(λ,
ˆ ) ∂x Ĝ00 (λ, , β)
(1 − ∂x J(,
+ K̂(λ, x)
ˆ )) − ∂x K̂(, λ) Jˆ(λ, )
(1 − K̂(λ, ))(1 − ∂x J(λ,
(5.14)
00
00
ˆ x) ∂x K̂(λ, ) Ĝ (λ, , β) − (1 − K̂(λ, )) ∂x Ĝ (λ, , β) .
+ J(λ,
ˆ )) − ∂x K̂(λ, ) Jˆ(λ, )
(1 − K̂(λ, ))(1 − ∂x J(λ,
(ii) For small > 0, this U rr satisfies (3.4) for any c > 0, and it is the Fourier-Laplace
transform of a Schwartz temperate distribution Prr
x [ω(t) ∈ db].
Proof. (i) Substitute for x in (5.13). Then differentiate both sides of (5.13) with respect to
x and then again substitute for x. As a result, we obtain the following equations,
U rr (λ, , β) = Ĝ00 (λ, , β) + K̂(λ, ) U rr (λ, , β)
ˆ ) ∂x U rr (λ, , β),
+J(λ,
∂x U rr (λ, , β) = ∂x Ĝ00 (λ, , β) + ∂x K̂(λ, ) U rr (λ, , β)
+∂x Jˆ(λ, ) ∂x U rr (λ, , β).
We can solve the equations for U rr (λ, , β) and ∂x U rr (λ, , β) and substitute the results into
(5.13). Then (5.14) easily follows. Part (ii) can be proved the same way as part (ii) of Lemma
5.2. 16
Theorem 5.12. In the sense of convergence of distributions, Prr
x [ω(t) ∈ db] converges to
Pr0
[ω(t)
∈
db]
defined
in
Theorem
5.5,
as
→
0.
x
Proof. If we let → 0 in (5.14) then we obtain
lim U rr (λ, x, β) = Ĝ0 (λ, x, β) − K̂(λ, x)
→0
∂x2 Ĝ0 (λ, 0, β)
∂x2 K̂(λ, 0)
.
The right hand side equals to U r0 in the proof of Lemma 5.4, so it is the Fourier-Laplace
transform of Pr0
x [ω(t) ∈ db].
6
Killing of monopole and reflection for dipole
Once again, we start with an approximating sequence. We will find a solution { P0r
x [ω(t) ∈ db] :
x ≥ 0, } to the equation
00
P0r
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da] P0r
+a [ω(t − s) ∈ db].
0
(6.1)
−∞
Remark 6.1. Intuitively speaking, we have the following model. Consider fixed > 0.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it is killed.
3. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it restarts from the point + ω(τ0 ).
Suppose { P0r
x [ω(t) ∈ db] : x ≥ 0, } solving (6.1) exists. We denote the Fourier-Laplace transform
0r
of Px [ω(t) ∈ db] by U 0r (λ, x, β). Then we apply the Fourier-Laplace transform to (6.1). Using
(3.13), we obtain for x > 0,
ˆ x) ∂x U r0 (λ, , β).
U 0r (λ, x, β) = Ĝ00 (λ, x, β) + J(λ,
(6.2)
The following lemma can be proved the same way as Lemma 5.11 so we omit its proof.
Lemma 6.2. (i) For > 0, the following is a solution of (6.2):
0
ˆ x) ∂x Ĝ (λ, , β) .
U 0r (λ, x, β) = Ĝ00 (λ, x, β) + J(λ,
ˆ )
1 − ∂x J(λ,
(6.3)
(ii) For small > 0, this U 0r satisfies (3.4) with any c > 0, and it is the Fourier-Laplace
transform of a Schwartz temperate distribution Por
x [ω(t) ∈ db].
Lemma 6.3. In the sense of convergence of distributions, P0r
x [ω(t) ∈ db] converges to a limit
as → 0.
17
Definition 6.4. We denote this limit by {P0r
x [ω(t) ∈ db] : x ≥ 0}, and call it the distribution
of BP P with the killing boundary for the monopole and the reflecting boundary for the dipole.
Proof of Lemma 6.3. We let tend to 0 in (6.3). Then (3.19) and (3.20) imply that
U 0r (λ, x, β) ≡ lim U 0r (λ, x, β)
→∞
2 00
ˆ x) ∂x Ĝ (λ, 0, β) .
= Ĝ00 (λ, x, β) − J(λ,
ˆ 0)
∂x2 J(λ,
(6.4)
When we take the principal value of λ1/4 , U 0r satisfies (3.4), and Lemma 3.1 implies our assertion.
ˆ 0) admits the usual inverse Fourier–Laplace transform. We
If <λ > 0, ∂x2 Ĝ00 (λ, 0, β)/∂x2 J(λ,
present the calculations. Let b ≥ 0. The residue theorem implies that
Z L
∂ 2 Ĝ00 λ, 0, β) 1
0r
I (λ, b) ≡
dβ exp{−iβb} − x
lim
ˆ 0)
2π L→∞ −L
∂x2 J(λ,
(6.5)
i
1/4
1/4
= √ exp{λ b θ2 } − exp{λ b θ2 } .
2 λ
For t > 0, we obtain that
Z 1+iM
1
0r
Q (t, b) ≡ −
dλ exp{λt} I 0r (λ, b)
lim
2πi M →∞ 1−iM
(6.6)
Z
2 ∞
4
=
dy exp{−y t} y sin(yb).
π 0
One can easily check that the density Q0r is a smooth bounded function of non-negative b if
t > 0, and it is integrable in t.
Theorem 6.5. For x ≥ 0, b ≥ 0, and t > 0, we define a function
Z t
p0r (t, x, b) ≡ p00 (t, x, b) +
ds J(s, x) Q0r (t − s, b),
(6.7)
0
where Q0r is the function in (6.6). Let {P0r
x [ω(t) ∈ db] : x ≥ 0} denote the distribution in
Definition 6.4. Then we have
0r
P0r
x [ω(t) ∈ db] = p (t, x, b) db,
(6.8)
and these distributions are continuous linear functionals on Bb [0, ∞).
Proof. The theorem is immediate from Lemma 6.2 and (6.6).
The following theorem is easily proved using (6.4) in the same way as Theorem 5.11 so we omit
its proof.
Theorem 6.6. (i) Let f be a bounded smooth function on the interval [0, ∞) such that f (0) =
0 = f 00 (0). We put
Z ∞
v(t, x) ≡
P0r
(6.9)
x [ω(t) ∈ db] f (b).
0
Then in the classical sense, this v satisfies (1.4) with ` = 0 and m = 2.
(ii) P0r
x [ω(t) ∈ db] satisfies Chapman-Kolmogorov equations (4.5), and the assertion in Remark
0
4.7 holds with P0r
x [w(t) ∈ db] in place of Px [ω(t) ∈ db].
18
7
Stopped monopole
7.1
Stopped monopole and dipole
Definition 7.1. Let {Pss
x [ω(t) ∈ db] : x ≥ 0} be defined by
00
Pss
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞n
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da]
0
−∞
o
+ Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da] δ(b − a) db.
(7.1)
Remark 7.2. The family of distributions given in the last definition represents a BP P whose
both particles are stopped after exiting (0, ∞). An intuitive description of the process is the
following.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. After BP P hits the interval (−∞, 0), both particles are stopped.
Theorem 7.3. For x ≥ 0, b ≥ 0, and t > 0, we define a linear functional on C1b [0, ∞),
Z t
ss
00
p (t, x, b) ≡ p (t, x, b) +
ds K(s, x) δ(b) − J(s, x) δ0 (b) .
(7.2)
0
Then we have
ss
Pss
x [ω(t) ∈ db] = p (t, x, b) db.
Both sides of the last formula are continuous linear functionals on
Proof. The result follows easily from (3.13) and Definition 7.1.
(7.3)
C1b [0, ∞).
Theorem 7.4. (i) Let f be a bounded smooth function defined on the interval [0, ∞) such that
f (4) (0) = 0 = f (5) (0). We define
Z ∞
v(t, x) ≡
Pss
(7.4)
x [ω(t) ∈ db] f (b).
0
Then in the classical sense, this v satisfies (1.4) with ` = 4 and m = 5.
(ii) Pss
x [ω(t) ∈ db] satisfies Chapman-Kolmogorov equations (4.5) in the sense of a linear functional on C1b [0, ∞).
ss
Proof. We apply the Fourier-Laplace transform to Pss
x [ω(t) ∈ db], and denote it by U (λ, x, β).
Then (7.2) is transformed into
1
ˆ x) iβ ,
+ J(λ,
(7.5)
λ
λ
from which we see that (5.10) holds for this U ss . Recalling (3.18), (3.19) and (3.20), we easily
see that
∂x4 U ss (λ, 0, β) = 0 and ∂x5 U ss (λ, 0, β) = 0.
U ss (λ, x, β) = Ĝ00 (λ, x, β) + K̂(λ, x)
This proves part (i) of the theorem, since the Fourier-Laplace transform is unique. For (ii), it is
sufficient to note that ∂x pss (t, x, b) is a linear functional on C1b [0, ∞) and the left hand side of
(4.5) is well-defined with p00 (t, x, b) replaced by ∂x pss (t, x, b). 19
7.2
Stopped monopole and killed dipole
Definition 7.5. Let {Ps0
x [ω(t) ∈ db] : x ≥ 0} be given by
00
Ps0
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z t Z ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da] δ(a − b) db.
s=0
(7.6)
a=−∞
Remark 7.6. A heuristic view of BP P with the boundary behavior specified in the section
title is this.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it is stopped at the point ω(τ0 ).
3. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it is killed.
Theorem 7.7. For x ≥ 0, b ≥ 0, and t > 0, we define a linear functional on Bb [0, ∞),
Z t
s0
0
p (t, x, b) ≡ p (t, x, b) +
ds K(s, x) δ(b).
(7.7)
0
We have
s0
Ps0
x [ω(t) ∈ db] = p (t, x, b) db,
(7.8)
where both expressions are continuous linear functionals on Bb [0, ∞).
Proof. The result follows easily from the definition (7.6).
Theorem 7.8. (i) Let f be a bounded smooth function on the interval [0, ∞) such that f 0 (0) =
0 = f (4) (0). We put
Z ∞
v(t, x) ≡
Ps0
(7.9)
x [ω(t) ∈ db] f (b).
0
Then in the classical sense, this v satisfies (1.4) with ` = 1 and m = 4.
(ii) Ps0
x [ω(t) ∈ db] satisfies Chapman-Kolmogorov equations (4.5) in the sense of a linear
functional on Bb [0, ∞).
Proof. Let U s0 (λ, x, β) be the Fourier-Laplace transform of Ps0
x . Apply the Fourier-Laplace
transform to (7.7). The equation is transformed to
U s0 (λ, x, β) = Ĝ00 (λ, x, β) + K̂(λ, x)
1
.
λ
(7.10)
From this, we see that (5.10) holds for this U s0 and that
∂x U s0 (λ, 0, β) = 0
and ∂x4 U s0 (λ, 0, β) = 0.
The first assertion of the theorem follows from this and from uniqueness of the Fourier-Laplace
transform. The proof of the second is the same as in Theorem 7.3. We present an example of a “not well-posed problem” for (1.4):
20
Example 7.9. Let f be a bounded smooth function on the interval [0, ∞) such that f 0 (0) =
0, f 00 (0) = 0, and f (4) (0) = 0. Recalling Theorems 5.7 and 7.8, we define functions
Z ∞
Z ∞
v1 (t, x) ≡
Pr0 [ω(t) ∈ db] f (b); v2 (t, x) ≡
Ps0 [ω(t) ∈ db] f (b).
o
o
These v1 and v2 are classical solutions of (1.4) with ` = 1. We consider their Laplace transforms
Z ∞
Vj (λ, x) ≡
dt exp{−λt} vj (t, x), j = 1, 2,
0
which satisfy
∂x4 Vj (λ, x) = −λVj (λ, x) + f (x),
j = 1, 2.
Since they are smooth and both satisfy ∂x Vj (λ, 0) = 0, we have
0 = ∂x5 Vj (λ, 0) = −λ∂x Vj (λ, 0) + f 0 (0) = 0,
j = 1, 2.
This fact implies that solutions to (1.4) are not unique when ` = 1 and m = 5. Note that an
analogous argument also holds when ` = 0 and m = 4.
7.3
Stopped monopole and reflected dipole
This case calls for an approximation procedure similar to those discussed in earlier sections. We
are looking for a family { Psr
x [ω(t) ∈ db] : x ≥ 0} satisfying
00
Psr
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da] δ(a − b) db
0
−∞
∞
Z tZ
+
0
−∞
(7.11)
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da] Psr
+a [ω(t − s) ∈ db].
Remark 7.10. An intuitive description of { Psr
x [ω(t) ∈ db] : x ≥ 0} is the following. Let be
a fixed positive number.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it is stopped at the point ω(τ0 ).
3. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it restarts from the point + ω(τ0 ).
Let U sr (λ, x, β) denote the Fourier-Laplace transform of Psr
x , if it exists. Recall (3.13) and
note that (7.11) can be transformed to
U sr (λ, x, β) = Ĝ00 (λ, x, β)+
1
ˆ x) ∂x U sr (λ, , β).
+ K̂(λ, x) + J(λ,
λ
(7.12)
The following lemma is similar to several lemmas presented earlier in the paper so we omit its
proof.
21
Lemma 7.11. (i) For > 0, the following is a solution of (7.12):
1
U sr (λ, x, β) = Ĝ00 (λ, x, β) + +K̂(λ, ) +
λ
00
ˆ ) ∂x Ĝ (λ, , β) + ∂x K̂(λ, )/λ .
+ J(λ,
ˆ )
1 − ∂x J(λ,
(7.13)
(ii) When > 0 is small, this U sr satisfies (3.4) for every c > 0 and it is the Fourier-Laplace
transform of a Schwartz temperate distribution Psr
x [ω(t) ∈ db].
Lemma 7.12. Psr
x [ω(t) ∈ db] converges to a limit as → 0, in the distribution sense.
Definition 7.13. The above limit will be denoted by {Psr
x [ω(t) ∈ db] : x ≥ 0}, and called the
distribution of BP P with stopped monopole and reflecting dipole on the boundary.
Proof. Recall (3.18) through (3.20), and let → 0 in (7.12). Then
1
U sr (λ, x, β) ≡ lim U sr (λ, x, β) = Ĝ00 (λ, x, β) + K̂(λ, x) −
→0
λ
2 00
2
ˆ x) ∂x Ĝ (λ, 0, β) + ∂x K̂(λ, 0)/λ .
− J(λ,
ˆ 0)
∂x2 J(λ,
This satisfies (3.4), so Lemma 3.1 implies our assertion.
(7.14)
We will apply the inverse Fourier-Laplace transform to
−
∂x2 Ĝ00 (λ, 0, β) + ∂x2 K̂(λ, 0)/λ
∂ 2 Ĝ00 (λ, 0, β)
∂ 2 K̂(λ, 0)
=− x
− x
ˆ 0)
ˆ 0)
∂x2 J(λ,
∂x2 J(λ,
λ ∂x2 Jˆ(λ, 0)
(7.15)
in the distribution sense. We already know a formula for the first term of (7.15), which is the
same as Q0r in (6.6). As for the second term
−
∂x2 K̂(λ, 0)
1
,
=√
2
ˆ
2 λ3/4
λ ∂x J(λ, 0)
we can apply the inverse Fourier-Laplace transform in the sense of Schwartz temperate distribution, to obtain
Z
2 ∞
−
dy exp{−y 4 t} δ(b).
π 0
We conclude that the inverse Fourier-Laplace transform of (7.15) is
Z
2 ∞
sr
0r
Q (t, b) ≡ Q (t, b) −
dy exp{−y 4 t} δ(b).
π 0
(7.16)
This is a linear functional on Bb [0, ∞). The following theorem is immediate from Lemma 7.12
and the above arguments. Therefore we omit its proof.
22
Theorem 7.14. For x ≥ 0, b ≥ 0, and t > 0, we define a linear functional on Bb [0, ∞),
sr
Z
00
t
p (t, x, b) ≡ p (t, x, b) +
ds K(s, x) δ(b)+
0
Z s
+
ds J(s, x) Qsr (t − s, b)
(7.17)
0
where Qsr is given in (7.16). We have
sr
Psr
x [ω(t) ∈ db] = p (t, x, b) db.
(7.18)
Both sides are continuous linear functionals on Bb [0, ∞).
Theorem 7.15. (i) Let f be a bounded smooth function on the interval [0, ∞) such that f 00 (0) =
0 = f (4) (0). We put
Z
v(t, x) ≡
∞
0
Psr
x [ω(t) ∈ db] f (b).
(7.19)
Then in the classical sense, this v satisfies (1.4) with ` = 2 and m = 4.
(ii) Psr
x [ω(t) ∈ db] satisfies Chapman-Kolmogorov equations (4.5), and Remark 4.7 holds with
sr
Px [ω(t) ∈ db] in place of P0x [ω(t) ∈ db].
Proof. Note that the Fourier-Laplace transform is unique. By (7.14), we see that (5.10) holds
for U sr . Moreover using (3.18) through (3.20), we have
∂x2 U sr (λ, 0, β) = 0 and ∂x4 U sr (λ, 0, β) = 0.
These prove (i), and (ii) can be proved by the same arguments as similar results in earlier
sections. 8
Stopped dipole
8.1
Stopped dipole and killed monopole
Definition 8.1. Let {P0s
x [ω(t) ∈ db] : x ≥ 0} be given by
00
P0s
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da]δ(a − b) db.
0
(8.1)
∞
Remark 8.2. Intuitively, we are dealing with the following model.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it is killed.
3. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it is stopped at the point ω(τ0 ).
23
The following theorem can be easily derived from (3.13).
Theorem 8.3. For x ≥ 0, b ≥ 0, and t > 0, we define a linear functional on C1b [0, ∞),
Z s
0s
00
p (t, x, b) ≡ p (t, x, b) +
ds J(s, x) δ0 (b).
(8.2)
0
Then we have
0s
P0s
x [ω(t) ∈ db] = p (t, x, b) db,
(8.3)
where both expressions are continuous linear functionals on C1b [0, ∞).
Theorem 8.4. (i) Let f be a bounded smooth function on the interval [0, ∞) such that f (0) =
0 = f (5) (0). We put
Z ∞
v(t, x) ≡
P0s
(8.4)
x [ω(t) ∈ db] f (b).
0
Then in the classical sense, this v satisfies (1.4) with ` = 0 and m = 5.
(ii) P0s
x satisfies Chapman-Kolmogorov equations (4.5).
Proof. If we denote by U 0s (x, λ, β) the Fourier-Laplace transform of P0s
x , then (8.3) is transformed into the following equality:
ˆ λ, β)
U 0s (x, λ, β) = Ĝ00 (x, λ, β) + J(x,
iβ
,
λ
(8.5)
from which we see that (5.10) holds for this U 0s . Using (3.18) through (3.20), we have
U 0s (λ, 0, β) = 0
and ∂x5 U 0s (λ, 0, β) = 0.
These prove (i). Note that ∂x p0s (t, x, b) is a linear functional on C1b [0, ∞). The second part of
the theorem can be proved by arguments used earlier in the paper. 8.2
Stopped dipole and reflected monopole
For the last time in this paper we use an approximation procedure to construct a process. The
first step is to find { Prs
x [ω(t) ∈ db] : x ≥ 0} such that
0
Prs
x [ω(t) ∈ db] = Px [ω(t) ∈ db]+
Z tZ ∞
+
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a monopole and in da] Prs
+ω(τ0 ) [ω(t) ∈ db]+
0
−∞
∞
Z tZ
+
0
−∞
(8.6)
Px [τ0 (ω) ∈ ds, ω(τ0 ) is a dipole and in da] δ(b − a)db.
Remark 8.5. The approximating process evolves according to the following rules. Fix > 0.
1. A monopole starts from a point x ≥ 0, and moves according to the transition probability
density p(t, x) until it hits the interval (−∞, 0).
2. If ω(τ0 ) is a monopole when BP P hits (−∞, 0), then it restarts from the point + ω(τ0 ).
24
3. If ω(τ0 ) is a dipole when BP P hits (−∞, 0), then it is stopped at the point ω(τ0 ).
Assuming existence, we denote by U rs (λ, x, β) the Fourier-Laplace transform of Prs
x . Recall
(3.13). We see that (8.6) is transformed into the following equation:
U rs (λ, x, β) = Ĝ00 (λ, x, β) + K̂(λ, x) U sr (λ, , β) + Jˆ(λ, x)
iβ
.
λ
(8.7)
We omit the proof of the following result.
Lemma 8.6. (i) For > 0, the following is a solution of (8.7):
ˆ x) iβ +
U rs (λ, x, β) = Ĝ00 (λ, x, β) + J(λ,
λ
00
ˆ )/λ
Ĝ (λ, , β) + iβ J(λ,
+ K̂(λ, x)
.
1 − K̂(λ, )
(8.8)
(ii) When > 0 is small, this U rs satisfies (3.4) for every c > 0 and it is the Fourier-Laplace
transform of a Schwartz temperate distribution Prs
x [ω(t) ∈ db].
We have just proved that if > 0, then there exist distributions Prs
x [ω(t) ∈ db] satisfying
equations (8.6). However the following proposition asserts that we cannot pass to 0 with .
Hence we cannot construct a BP P with reflection for the monopole and stopped dipole, at least
not using our method.
Proposition 8.7. (8.8) diverges as → 0.
Remark 8.8. Proposition 8.7 is not very surprising if we consider the corresponding problem
(1.4). Table 1.2 suggests that, if a BP P with reflection for the monopole and stopped dipole
exists, then it corresponds to (1.4) with ` = 1 and m = 5. But solutions to this boundary value
problem (1.4) are not unique. So we cannot find a “fundamental solution” of such (1.4), which
would serve as a transition density for the corresponding BP P .
Proof of Proposition 8.7. From (3.18) through (3.20), we see that
2 2
∂x K̂(λ, 0) + o(2 )
2 √
2 λ
=−
+ o(2 ),
2
1 − K̂(λ, ) =
Ĝ00 (λ, , β) +
This shows that
iβ Jˆ(λ, )
iβ
ˆ 0) + o()
=
× ∂x J(λ,
λ
λ
iβ
=
+ o().
λ
Ĝ00 (λ, , β) + iβ Jˆ(λ, )/λ
1 − K̂(λ, )
diverges as → 0, and so (8.8) also diverges.
25
(8.9)
(8.10)
9
Conservation of charge
Let {Qx [ω(t)] : x ≥ 0} be the generic notation for the distribution of BP P on [0, ∞) with any
boundary conditions for the monopole and dipole.
Definition 9.1. We say that Qx [ω(t) ∈ db] is conservative if
Z
ρ(t, x) ≡
Qx [ω(t) ∈ db] = 1 for all t > 0 and x > 0.
(9.1)
[0,∞)
Theorem 9.2. Let {Qx [ω(t) ∈ db] : x ≥ 0} be the distribution of any BP P on [0, ∞) constructed
in this paper. Then Qx [ω(t) ∈ db] is conservative if and only if there is no killing for the
monopole.
Remark 9.3. This result intuitively supports for the names “monopole” and “dipole.” Since
a dipole carries two charges of equal magnitude but opposite signs, its loss does not change the
total amount of charge. On the other hand, the monopole carries charge of single sign, and its
loss changes the total amount of charge.
Proof of Theorem 9.2. Inspecting Table 1.2, we see that there is no killing for the monopole if
and only if ` ≥ 1 and m ≥ 1.
Assume that ` ≥ 1 and m ≥ 1 in (1.4). Since ρ(0, x) = 1 in view of (9.1), we see that ρ is a
solution of (1.4) with the constant initial function
f (x) ≡ 1.
The constant function v(t, x) ≡ 1 is a solution of (1.4) with f ≡ 1, if ` ≥ 1 and m ≥ 1. Now
uniqueness of the solution implies that ρ(t, x) ≡ 1, and Qx [ω(t) ∈ db] is conservative.
On the other hand, if ` = 0, then the constant v(t, x) ≡ 1 is not a solution of (1.4) with the
constant initial function f ≡ 1, and uniqueness of the solution implies that ρ(t, x) 6= 1 for some
(t, x). In this case, Qx [ω(t) ∈ db] is not conservative. References
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